Coded Caching for Multi-level Popularity and Access

Transcription

1 Coded Caching for Multi-level Popularity and Access Jad Hachem, Student Member, IEEE, Nikhil Karamchandani, Member, IEEE, and Suhas Diggavi, Fellow, IEEE Abstract arxiv: v2 [csit] 3 Dec 205 To address the exponentially rising demand for wireless content, use of caching is emerging as a potential solution It has been recently established that joint design of content delivery and storage (coded caching) can significantly improve performance over conventional caching Coded caching is well suited to emerging heterogeneous wireless architectures which consist of a dense deployment of local-coverage wireless access points (APs) with high data rates, along with sparsely-distributed, large-coverage macro-cell base stations (BS) This enables design of coded caching-and-delivery schemes that equip APs with storage, and place content in them in a way that creates coded-multicast opportunities for combining with macro-cell broadcast to satisfy users even with different demands Such coded-caching schemes have been shown to be order-optimal with respect to the BS transmission rate, for a system with single-level content, ie, one where all content is uniformly popular In this work, we consider a system with non-uniform popularity content which is divided into multiple levels, based on varying degrees of popularity The main contribution of this work is the derivation of an order-optimal scheme which judiciously shares cache memory among files with different popularities To show order-optimality we derive new information-theoretic lower bounds, which use a sliding-window entropy inequality, effectively creating a non-cutset bound We also extend the ideas to when users can access multiple caches along with the broadcast Finally we consider two extreme cases of user distribution across caches for the multi-level popularity model: a single user per cache (single-user setup) versus a large number of users per cache (multi-user setup), and demonstrate a dichotomy in the order-optimal strategies for these two extreme cases I INTRODUCTION Broadband data consumption has witnessed a tremendous growth over the past few years, due in large part to multi-media applications such as Video-on-Demand This increased demand has been managed in the wired internet via Content Distribution Networks (CDNs), by mirroring data in various locations and in effect pushing the content closer to the end users Wireless data consumption, driven by the increased demand for high-definition content on mobile devices, has also grown at a significant rate [] and is testing the limits of our underlying wireless communication systems [2] However, simply borrowing the CDN solution from wired networks and applying it to wireless systems is insufficient to solve the wireless content delivery problem In the wired Internet, CDNs remove the bottleneck at the content distribution server by utilizing repeated demand of particular content It has the most gains when the local communication link is not the bottleneck [3] In wireless cellular usage, this is typically not true as the (cellular) wireless hop is a bottleneck link The broadcast nature of wireless can be used as an advantage to alleviate this problem This, along with the emerging heterogeneous wireless network can be used to provide an architecture for wireless content distribution The heterogeneous wireless network (HetNet) architecture emerging for 5G consists of a dense deployment of wireless access points (APs) with small coverage and relatively large data rates, in combination with cellular base-stations (BS) with large coverage and smaller data rates For example, the access points could be WiFi or emerging small-cells (or femto-cells), which provide high data rate for short ranges The consequence of this emerging architecture is that a user could potentially receive broadcast from the BS as well as connect to (several) wireless APs Therefore, we could place caches at local APs and complement them with macro-cellular (BS) broadcast In this paper we study a problem based on an architecture where content is stored at multiple APs without a priori knowing the user requests, and the base-station broadcast is used judiciously to complement the local caching, after the user requests are known This is motivated by the new approach initiated in [4], [5] where it has been shown that joint design of storage and delivery (aka coded caching ) can significantly improve content delivery rate requirements This was enabled by content placement that creates (network-coded) multicast opportunities among users with access to different storage units, even when they have different (and a priori unknown) requests This enables an examination of the optimal trade-off between cache memory size and broadcast delivery rate The setup studied in [4], [5] consisted of single-level content, ie, every file in the system is uniformly demanded However, it is well understood that content demand is non-uniform in practice, with some files being more popular than others Motivated by this, [6], [7], [8], [9], [0], [] considered such non-uniform content demand, following different models In [6], [7], [8], the setup considered a single user per cache requesting a file independently and randomly according to some (arbitrary) probability distribution that represents content popularity These works studied the trade-off between the average rate and the cache memory A memory-sharing scheme was proposed in [6], and its achievable rate was characterized However, from our J Hachem and S Diggavi are with the Department of Electrical Engineering, University of California, Los Angeles N Karamchandani is with the Department of Electrical Engineering, Indian Insitute of Technology, Bombay This work was supported in part by NSF grant #42327 and a gift by Qualcomm Inc

2 2 understanding, this scheme was not shown to be order-optimal in general In [7], a different scheme was proposed, based on a clustering of the most popular files into a single content level, which was shown to be order-optimal for Zipf-distributed content and, more recently, for arbitrary distributions in [8] By contrast, in [9], a deterministic multi-level popularity model was introduced (simultaneous to the aforementioned other non-uniform popularity models), where content is divided into discrete levels based on popularity In this paper, we focus on this model and study it mostly in the context where a large number of users connect to each cache ( multi-user model ), and, for each level, a fixed and a priori known fraction of the users per cache request files from said level It is easy to see that, when the number of users per cache is large enough, this deterministic model will closely approximate an equivalent stochastic-demands model similar to [6], [7], [8] We will also study the scenario where users could connect to multiple access points (caches) as well as listen to the broadcast to get the desired content In short, the setup considered has a different popularity model and user population as well as cache access than considered in earlier literature We will also compare the results, for this popularity model, between setups with many users per cache (multi-user setup) and a single user per cache (single-user setup) The main contribution of this work is, for any given multi-level content popularity profile, to approximately solve the tradeoff of the transmission cost at the BS with the storage cost at the APs In addition, we also approximately solve the case where users have access to multiple APs Finally, we study the effect of number of users per cache in the multi-level content popularity model In particular, the following are the core technical contributions of the work: 2 Non-cut-set-based information-theoretic lower bounds, for the multi-user model with users accessing multiple caches, which are used to evaluate the performance of the proposed schemes A memory-sharing scheme, which divides the available storage at each cache (AP) among the various popularity levels A striking aspect of this (order-optimal, for the multi-user setup) solution is that, in some regimes, it is better to store some less popular content without completely storing the more popular content, even when cache memory is available We demonstrate order-optimality of the scheme with respect to the information-theoretic lower bound that is independent of the number of popularity levels, number of users, files, and caches We demonstrate that drastically different strategies are order-optimal for the multi-user and single-user setup In the singleuser case, we show that clustering the most popular levels and giving them all the memory, leaving none for the rest, is order-optimal; a strategy proposed in [7], [8] In contrast, the multi-user case requires a complete separation of the different levels and a division of the memory between them The paper is organized as follows Section II formulates the problem, describing precisely the multi-user and single-user setups We establish some background in Section III, which enables us to state the main results in Section IV The caching and delivery strategy for the multi-user set up, as well as corresponding lower bounds, are given in Section V, while the single-user setup is studied in Section VI A brief discussion about the dichotomy in the two setups is given in Section VII The paper concludes in Section VIII with a discussion and some numerical evaluations to interpret the results Many of the detailed proofs are given in the appendices A Related work Content caching has a rich history and has been studied extensively, see for example [2] and references therein More recently, it has been studied in the context of Video-on-Demand systems where efficient content placement and delivery schemes have been proposed in [3], [3], [4], [5] The impact of content popularity distributions on caching schemes has also been widely investigated, see for example [6], [7], [8] Most of the literature has focused on wired networks and, as argued before, the solutions there do not carry directly to wireless networks Recently, [9] proposed a caching architecture for heterogeneous wireless networks, with the small-cell or WiFi access points acting as helpers by storing part of the content A content placement scheme is formulated and posed as a linear program However, the (information-theoretic) optimality of such schemes was not examined in that work Another aspect (also common to most of the papers in the content caching literature) is that the delivery phase used independent unicasts to serve the different users The important observation to utilize broadcast to improve system performance by serving multiple users simultaneously was made in [5], [4] They initiated the study of coded caching where joint design of storage and delivery was considered for the case with a single level of files and single cache access during delivery by proposing an order-optimal coded caching scheme These results have been extended to online caching systems in [20], heterogeneous cache sizes [2], unequal file sizes [22], and improved converse arguments [23], [24] Efficient coded caching schemes have been devised in [25], and the effect of finite file sizes has been investigated in [26] Content caching and delivery has also been studied for hierarchical tree topologies [27], [28], device to device networks [29], [30], multi-server topologies [3], and heterogeneous wireless networks [0] We refer to an order-optimal result as one that is within a constant multiplicative factor from the information-theoretic optimum The constant is to be independent of the number of users, caches, memory size and number of popularity levels 2 Shorter versions of these results have been published in [9], [0], []

3 3 0-2 empirical popularities Zipf approximation 0-3 popularities files Fig Empirical popularities of some YouTube videos (based on number of views), with an approximating Zipf distribution Coded caching was extended to non-uniform popularity models in [6], [7], [8], [32], where the setup considered a single user per cache requesting a file independently and randomly according to some (arbitrary) probability distribution that represents content popularity The trade-off between memory and average delivery rate was studied in these works Our work differs from these as it uses a deterministic multi-level popularity model introduced in [9], enabling a worst-case rather than average case analysis We analytically characterize the order-optimal splitting parameters for the memory-sharing scheme, even with user access to multiple caches The dichotomy of order-optimal schemes between having multiple users per cache and a single user per cache is also demonstrated for this multi-level popularity model Other related work includes [33] which derives scaling laws for content replication in multihop wireless networks; [34] which explores distributed caching in mobile networks using device-to-device communications; [35] which studies the benefit of coded caching when the caches are distributed randomly; and [36] which explores the benefits of adaptive content placement, using knowledge of user requests A Setup II SETUP, NOTATION, AND FORMULATION Consider a network where a group of users request files from a server All files are assumed to be of size F bits Prior to any user requests, a placement phase occurs in which information about these files is placed in the caches of K access points (APs); each cache has a capacity of MF bits Then, in the delivery phase, users connect to the different caches, and each requests a file based on an underlying file popularity model: more popular files are more likely to be requested The server then sends, through the base station (BS), a broadcast message of size RF bits that all the users can hear The users combine the broadcast with the contents of their cache to recover the file that they have requested Clearly, there is a trade-off between the values of M (the cache memory ) and R (the broadcast rate ): the larger the cache memory, the more information the caches can store, and hence the smaller the broadcast rate needed to serve the requests Our goal is to characterize this trade-off B Multi-level popularity In multi-media applications such as video-on-demand, we often find that a small number of files are requested by many more users than the rest of the files This difference in popularity can easily influence the caching system described above For example, when deciding what to store in the (limited-capacity) caches, one would want to give more of the cache memory to the more popular files, since they will be, on average, requested more often Different popularity models have been considered in the literature The simplest model was studied in [4], [5] In this model, all files are equally popular, meaning that there is no preference among the users to choose one file over the others The results were of a worst-case nature, ie, they are true for all possible (valid) combinations of user demands While this model allowed the first approximate-optimality caching result and introduced the idea of coded caching, it is not an accurate representation of typical multi-media data To introduce the effect of file popularity, we studied a multi-level popularity model in [9], [0], [], in which files are divided into different popularity classes (levels), and files within each class are equally popular The results are also based on a worst-case analysis Finally, probabilistic models were also studied in the literature In these, user demands are stochastic and follow some probability distribution, and the focus is on average results rather than worst-case ones There has been focus on Zipf distributions [6], [7], which can be seen in examples such as YouTube videos (see Fig based on data from [37]), but also on arbitrary distributions [6], [8]

4 4 BS N N 2 AP AP2 AP3 AP4 Fig 2 Multi-user setup with K = 4 caches, and L = 2 levels with (U, U 2 ) = (2, ) users per cache Both levels have an access degree of The popularity model that we consider in this paper is the multi-level model The files are divided into L popularity levels, such that all files in a single level are equally popular 3 The levels consist of N,, N L files Furthermore, the total number of users in the system who are requesting files from each level i is fixed and known to the designer a priori To motivate the determinism in this last point, consider the following example Suppose there are two popularity levels, and assume a stochastic-demands setup where each user is three times as likely to request a file from the first level as he is from the second If there are 40 users in the network, then we would expect that about 30 of them will request a file from the first level, and 0 from the second By the law of large numbers, when a large number of users is present in the system, we expect a concentration of the number of users requesting files from each level around their means Because of this concentration, the stochastic-demands model will closely resemble the determinism in the multi-level model that we adopt In this paper, we will use the phrase user profile to refer to the arrangement of users across the caches Specifically, it is the number of users requesting a file from level i at cache k, for every pair (i, k) Moreover, we define the user demand vector or user request vector as the vector of specific files requested by each user C Multi-level access Another aspect of the caching problem that we study is the possibility of users to access more than one cache This introduces a third dimension to the trade-off The more caches users can access, the more information they can gather from them for the same memory, and hence the less information they need from the broadcast transmission However, an increased degree can be costly to users who must now establish connections with multiple APs We introduce this multiple-access dimension to the problem by augmenting the multi-level popularity model with a multilevel access aspect Users are required to access a certain number of caches based on the popularity level of the file they request Specifically, all users demanding a file from level i must access d i caches We call d i the access degree of level i Because we would prefer that users connect to the nearest caches, these d i caches are consecutive, with a cyclic wrap-around for symmetry, as in Fig 3 D Number of users As we will see later, the caching system that we have described behaves differently depending on the number of users per cache In particular, it requires different strategies when this number is large or small We focus our attention on the two extreme cases of number of users In one extreme, we assume that there is only one user at every cache Recall from Section II-B that the popularity model assumes the number of users requesting a file from each level is fixed and known However, since there is only one user per cache, we do not know beforehand which user will connect to which cache Thus the user profile is unknown We call this extreme the single-user setup In the other extreme, the number of users per cache is large enough that the concentration of the levels requested by each user manifests itself, not only on the overall set of users, but also on the set of users connected to a single cache In other words, we assume that the number of users requesting files from each level is fixed per cache Furthermore, we assume a symmetry across the caches, so that every level is represented equally at each cache We call this setup the multi-user setup After introducing the two setups, we now give their formal definitions ) Multi-user setup: Consider the setup shown in Fig 3 For every level i and every consecutive subset of d i caches, there are exactly U i users connecting to these caches and requesting a file from level i Notice that every level is represented equally at every cache

5 5 BS N N2 AP AP2 AP3 AP4 Fig 3 Multi-user setup with K = 4 caches, and L = 2 levels with (U, U 2 ) = (2, ) users per cache, and access degrees of (d, d 2 ) = (, 2) BS N N 2 AP AP2 AP3 AP4 Fig 4 Single-user setup with K = 4 caches, and L = 2 levels with (K, K 2 ) = (3, ) users 2) Single-user setup: Consider now the setup in Fig 4, depicting the other extreme We have only one user connecting to every cache, for a total of K users The only information known a priori is that, for each level i, exactly K i out of the K users will request a file from i However, we do not know which users these will be For symmetry, we assume only single-access in the single-user setup, ie, d i = for all levels i E Rate-memory trade-off Recall that our main goal is to characterize the trade-off between the broadcast rate R and the cache memory M The determinism in the setups allows us to study worst-case trade-offs as opposed to average-case trade-offs We say that a pair (R, M) is achievable if there exists a placement-and-delivery strategy that uses caches of memory M and transmits, for any possible combination of user requests (valid within the popularity model), a broadcast message of rate at most R that satisfies all said requests with vanishing probability as the file size F grows Our goal is to find all such achievable pairs In particular, we wish to find the optimal rate-memory trade-off: where the minimization is done over all possible strategies R (M) = inf {R : (R, M) is achievable}, F Problem formulation The problem of finding an exact characterization of the rate-memory trade-off is difficult even for the simplest cases [4] Therefore, in this paper, we will instead consider approximate characterizations In particular, we wish to find an achievability strategy that results in a rate-memory trade-off R(M) such that: cr(m) R (M) R(M), where c is some constant We say that such a strategy is order-optimal We allow c to depend on only one parameter: the maximum access degree D = max i d i In practice, we do not expect that one user will be required to access a large number of caches, and so D would be quite small However, c must be independent of all other parameters 3 For a discussion of the multi-level popularity model, see Section VIII-A

6 6 TABLE I NOTATION All setups K # of caches L # of popularity levels N i # of files in level i F file size R broadcast rate (normalized) M cache memory (normalized) R (M) optimal rate-memory trade-off Multi-user setup U i # of users per cache for level i d i access degree of level i D maximum access degree β = level-separation factor 98 Single-user setup K i total # of users for level i G Regularity conditions We assume the following two regularity conditions First, for every popularity level i, there are more files than users In the multi-user setup, this means: i, N i KU i () In the single-user setup, we would write: i, N i K i (2) This can be seen, for example, in video applications such as Netflix, where files would be video segments of a few seconds to a few minutes To borrow an example from [27], if a database has 000 popular movies of length 00 minutes, and each movie is divided into files (segments) of one minute each, the result is 00,000 files It is unlikely that over 00,000 users will each be watching one of those 000 movies at the same time Second, in the multi-user setup only, we assume that no two levels have very similar popularities The popularity of a level can be written as the number of users per file of the level, ie, as U i /N i Hence, if i is a more popular level than j, the regularity condition states: 4 U i /N i D = 98D, (3) U j /N j β where β = 98 is called the level-separation factor The reasoning behind this condition is that, if it did not hold for some levels i and j, then we can think of them as essentially one level with N i + N j files and U i + U j users per cache The resulting popularity Ui+Uj N i+n j would be close to both U i /N i and U j /N j H Notation table For reference, we present in TABLE I all the notation that we use in this paper III PRELIMINARIES Traditional caching only uses multiple-unicast transmissions from the server to the users As a result, the total transmission size was proportional to the number of users, for any value of the cache memory Coded caching, initially introduced in [4], brought a drastic improvement by eliminating the dependence of the transmission size on the number of users (except for very small cache memory) This technique was shown to be approximately optimal in [4] (a centralized version) and in [5] (a decentralized version), under a setup with a single level of popularity and a single user at every cache, with a single-access structure We will refer to this setup as the Basic Setup, because it will form the basis of our main analysis The scheme for the Basic Setup, in its decentralized form (on which will we henceforth focus) consists in placing a random sampling of bits from all files in every cache, independently Consequently, there will be some overlap, but also some differences, in the bits present in every cache The BS then transmits linear combinations of these bits, taking advantage of the overlaps as well as the differences, so that the same linear combination can be useful for multiple users at once The resulting rate-memory trade-off is given in the following Lemma 4 As we will see in later sections, the variables U i and N i will often appear inside square roots For this reason, phrasing the regularity condition using the square roots is more useful

7 7 BS N AP AP2 AP3 AP4 Fig 5 Generalized single-level setup with U = 3 users per cache, and an access degree d = 2 BS N AP AP2 AP3 AP Fig 6 An illustration of the scheme used on the example from Fig 5 Caches are colored into d = 2 colors, and the files are divided and colored with the same colors In parallel, the users are divided into du = 6 groups, where users fro m the same group have no overlapping caches Lemma (Rate for the Basic Setup [4]) For a single-level caching system with K caches, N files, a single user per cache with an access degree of, and a cache memory of M, the following rate is achievable: { } ( N R 0 (M, K, N) = min M, K M ) N Furthermore, this rate is within a constant of the optimum Notice that, when M > N/K, then the rate becomes (N/M ), removing all dependence on the number of users Under traditional caching, the rate would have been K( M/N) A Generalizing to multi-user, multi-access In the multi-level setups, every level can have more than one user per cache, and can also have an access degree that is larger than Thus, the first step towards solving the multi-level problem is to generalize the single-level problem to one where these two parameters are no longer Fig 5 illustrates such a setup We adopt the following strategy We first color all the caches into d colors, such that every user is connected to exactly one cache of every color In parallel, we split every file into d equal subfiles, and color each subfile using the same colors as the caches Next, we group all the KU users into du groups of K/d users each, such that no two users in the same group share any cache The end result is illustrated in Fig 6 In the placement phase, we consider each of the d colors separately The caches of a certain color perform a simple placement on the subfiles of the same color Since each file is split into d equal subfiles, every cache can hold dm subfiles, the equivalent of M files In the delivery phase, each of the du groups is considered separately For every group and every color, we have a subsystem where K/d users are each requesting a subfile of size F/d bits from one like-colored cache of size dm subfiles The total broadcast size is thus: RF = du d R 0 (dm, K/d, N) (F/d) bits Note that the memory given to the R 0 function is dm This is because the files in each subsystem have a size of F/d bits, and thus the total cache size of MF bits must be normalized by F/d bits Using the above equation with Lemma, we get the following theorem Theorem (Single-level rate-memory trade-off) Given a single-level caching system, with N files, K caches, U users at each cache with access degree d, and a cache memory of M, the following rate is achievable: { } ( N R SL (M, K, N, U, d) = U min M, K dm ) (4) N

8 8 Popular files Less popular files W W2 W 2 W2 2 W 2 N2 BS X r Z Z 2 AP AP2 user user 3 user 2 Ŵ r Ŵ 2 r3 Ŵ r2 Fig 7 Small example that illustrates multi-level popularity and access R N2 2 M Fig 8 Optimal rate-memory trade-off R (M) for the small example While, in the single-access case, a cache memory M < N meant a non-zero transmission rate, adding the multi-access aspect results in a zero achievable rate for the smaller memory value of N/d Intuitively, at M = N/d, it is as though we apply an erasure-correcting code on all the files and spread it across the caches, such that any d caches can reproduce all files The results presented in this section will be key to our solution of the multi-level caching problem Indeed, in all that will follow, we use the above-described (decentralized) coded caching scheme as a black box that gives a rate R SL (M, K, N, U, d) for input parameters M, K, N, U, and d B A small multi-level example with exact characterization In order to illustrate the general multi-level problem, we will here present a small example that combines both multi-level popularity and multi-level access We give, for this example, an exact characterization of the rate-memory trade-off Consider the setup in Fig 7 The server holds files from L = 2 popularity levels The first level has N = 2 files, and the second level has N 2 4 files There are two APs, each equipped with a cache of memory (ie, size normalized by file size) M There is one user accessing each cache and requesting a file from level (users and 2), and a third user accessing both caches and requesting a file from level 2 (user 3) Theorem 2 (Exact characterization for the small example) For the setup shown in Fig 7, the optimal rate-memory trade-off is plotted in Fig 8 and is characterized by: { R (M) = max 3 2M, 5 2 M, 2 2 M, M 2 } N 2 /2 The proof of Theorem 2 is given in Appendix C However, for illustration and to gain intuition about the general problem, we will here briefly discuss the achievability scheme for two values of the cache memory M: M = (for which R = 3/2 is achievable) and M = /2 (for which R = 2 is achievable) Suppose that M =, so that each cache can hold the equivalent of one file We first split each file in level into two equal parts: Wn = (Wn,a, Wn,b ), for n =, 2 Now, each cache exclusively stores one half of each popular file, and stores nothing

9 9 from level 2 Thus the first cache will contain (W,a, W2,a) and the second cache will contain (W,b, W 2,b ) When the users make their requests, the BS transmits Wr 2 3 completely for user 3, and sends a coded transmission (Wr W,b r 2,a) for users and 2 together Combining the transmission with the contents of their respective caches, each of users and 2 can recover the file that they have requested In total, the BS would have transmitted one complete file (Wr 2 3 ), plus the equivalent of one half-file (the linear combination), for a total rate of R = 3/2 Suppose now that M = /2, ie, each cache can only hold the equivalent of half a file We again split the two level- files just like in the previous case However, this time the first cache stores (W,a W2,a) while the second cache stores (W,b W 2,b ) When the users make their requests, the BS again transmits W r 2 3 to serve user 3, but also sends Wr and,b Wr 2,a for users and 2 This allows them to recover the file that they have requested by combining the transmission with the side-information available at their caches Since the BS has transmitted one complete file and two half-files, the total rate is R = 2 While, in this small example, an exact characterization of the rate-memory trade-off was found, this is difficult in general For this reason, we focus our attention on order-optimality results as stated in Section II-F IV MAIN RESULTS In this section, we provide the approximately optimal rate-memory trade-off for each of the two setups (multi-user and single-user) We discuss how each such trade-off is achieved A Multi-user setup The placement-and-delivery strategy that we adopt for the multi-user setup is a memory-sharing strategy It consists of dividing the cache memory between all the L levels, and then treating each level as a separate caching sub-system, with the reduced memory In other words, we give level i a memory α i M, where α i [0, ] and i α i =, and we then apply a single-level placement-and-delivery strategy for this level on this α i M memory, separately from the other levels The total rate for this scheme is thus: L R MU (M, K, {N i, U i, d i } i ) = R SL (α i M, K, N i, U i, d i ), (5) where R SL ( ) is defined in (4) By optimizing the overall rate over the memory-sharing parameters {α i } i, we establish a memory allocation which we will show is order-optimal At a high level, this allocation is done by partitioning the popularity levels into three sets: H, I, and J The levels in H have such a small popularity that they will get no cache memory On the opposite end of the spectrum, the most popular levels are assigned to J and are given enough cache memory to completely store all their files in every cache Finally, the rest of the levels, in the set I, will share the remaining memory among themselves, obtaining some non-zero amount of memory but not enough to store all of their files Our choice of the (H, I, J) partition and corresponding memory assignments are discussed in Section V-A This choice results in the following achievable rate Theorem 3 Given a multi-user caching setup, with K caches, L levels, and, for each level i, N i files and U i users per cache with access degree d i, and a cache memory of M, the following rate 5 is achievable: R MU (M) ( ) 2 i I Ni U i KU h + M h H j J N d i U i, j/d j i I where (H, I, J) is a particular type of partition of the set of levels called an M-feasible partition 6 Intuitively, since a level h H receives no cache memory, all requests from its KU h users must be handled directly from the broadcast Since, by regularity condition (), we have N i KU i for all levels i, then in the worst case a total of KU h distinct files must be completely transmitted for the users requesting files from level h The users in set J require no transmission as the files are completely stored in all the caches; however, it does affect the rate through the memory available for levels in I This is apparent in the expression M j J N j/d j Finally, the levels in I, having received some memory, require a rate that is inversely proportional to the effective memory and that depends on the level-specific parameters N i, U i, and d i The structure of the (H, I, J) partition that we have chosen allows us to efficiently compute it for all values of the cache memory M Indeed, we provide an algorithm in Section V-A that can find this partition, as well as the corresponding memorysharing parameters α i, for every M in Θ(L 2 ) running time Briefly, as M is increased, levels get promoted from the set H to I to J The sequence of these promotions is directly determined by the popularity of the levels 5 This expression of the rate is a slight approximation that we use here for simplicity as it is more intuitive An exact and complete description of the achievable rate is given in Section V-A 6 This type of partition is defined in Definition and elaborated upon in Section V-A i=

10 0 We now discuss the order-optimality of the memory-sharing scheme in the multi-user setup We develop new, non-cut-set lower bounds on the optimal rate, which use sliding-window entropy inequalities [38], and show that the scheme achieves a rate that is within a constant factor of the optimal Note that this constant is independent of all the problem parameters except the largest AP access degree D Theorem 4 For all valid values of the problem parameters K, L, {N i, U i, d i } i, and M, we have: R MU (M) R (M) cd, where R MU (M) is the rate achieved by memory-sharing, R (M) is the optimal rate over all strategies, D is the largest AP access degree D = max i d i, and c = 9909 is a constant (independent of all problem parameters) The gap between the rate achieved by the memory-sharing strategy and the optimal rate is linear in D As we have argued earlier, we would not expect a situation where one user connects to too large a number of APs, and so D can be thought of as a constant The lower bounds needed to prove Theorem 4 have to include the effect of all the popularity levels on the transmission rate However, these effects can be very different, especially when some files are much more popular than others, something that cut-set bounds alone cannot account for Using sliding-window subset entropy inequalities [38], we can combine multiple cutset bounds that correspond to the different levels, without making any assumptions on the achievability scheme The resulting bounds bring out the necessity for memory-sharing Furthermore, the constant c in Theorem 4 is rather loose so as to simplify the analysis Numerics show that, in practice, this constant is much smaller For example, if we have K = 20 caches, L = 3 popularity levels consisting of (N, N 2, N 3 ) = (200, , ) files, (U, U 2, U 3 ) = (0, 5, ) users per cache, and access degrees of (d, d 2, d 3 ) = (,, ), then the gap is less than 68 B Single-user setup In the single-user setup, the scheme that we propose is quite different Instead of separating the levels, we cluster a subset of them into a super-level that will be treated as essentially one level Specifically, we partition the levels into two subsets: H and I The set I will be clustered into one super-level, and all of the memory M will be given to it, while H will be given no memory To understand how to choose H and I, consider the following rough analysis Suppose that all levels except one (let s call it j) have been split into H and I Then, the rate, using Theorem, would be: R = R SL (0, h H K h, h H N h,, ) + R SL (M, i I K i, i I N i,, ) i I K h + N i M h H If we were to add level j to H, that would result in the addition of a K j term, since all K j requests would be completely served by the broadcast On the other hand, if it is added to I, then we would get an additional N j /M term, since the total number of files in I would increase by N j Clearly, it is beneficial to choose the smaller of the two quantities Though the above analysis is rough, its main idea still holds In general, we choose the partition (H, I ) as follows: H = { h {,, L} : M < N h K h } ; I = (H ) c (6) Then, by giving all of the memory to I, we can apply a single-level caching-and-delivery scheme to obtain the rate in the following theorem Theorem 5 Consider the multi-level, single-user setup with L levels, N i files and K i users for each level i, and cache memory M Then, the following rate is achievable: R SU (M) = { i I K h + max N } i M, 0, h H where H and I are as defined in (6) This scheme turns out to be order-optimal, as we state in the next theorem Theorem 6 The rate achieved by the clustering strategy in Theorem 5 is within a constant multiplicative factor of the information-theoretic optimum: R SU (M) R (M) 72,

11 broadcast transmission rate memory-sharing scheme traditional LFU cache memory Fig 9 Comparison of the memory-sharing scheme with traditional LFU, in the multi-user setup broadcast transmission rate clustering scheme traditional LFU cache memory Fig 0 Comparison of the clustering scheme with traditional LFU, in the single-user setup where R SU (M) is the rate achieved by clustering, and R (M) is the information-theoretically optimal rate Unlike in the multi-user case, cut-set bounds are sufficient to show order-optimality in this case Indeed, a single cut-set bound allows us to capture the fact that the user profile (ie, the level of the file requested at each cache, as defined in Section II-B) is not determined beforehand At the same time, it brings out the necessity of clustering levels by mixing their demands As before, however, these bounds do not make any assumptions on the achievability strategy The scheme suggested in Theorem 5 is similar to the results in [7] for Zipf popularity distributions and in [8] for arbitrary distributions However, this is done for the multi-level popularity model, and Theorem 6 establishes a universal approximation for worst case rate, rather than average rate C Comparison with LFU One of the most common caching schemes conventionally in use is the Least-Frequently-Used (LFU) scheme This strategy stores the most popular files only, as many as the caches can hold In traditional LFU, no coding is done in the delivery phase We give an example for each of the two setups, to show how the respectively chosen schemes are superior to LFU in each context ) Multi-user example: Consider two levels (L = 2), with K = 30, (N, N 2 ) = (600, 000), (U, U 2 ) = (20, 0), and (d, d 2 ) = (, ) The rates achieved by the memory-sharing strategy and traditional LFU are plotted against memory M in Fig 9 Memory-sharing performs up to 29 times better than LFU in this example 2) Single-user example: Consider two levels (L = 2), with (N, N 2 ) = (500, 000) and (K, K 2 ) = (30, 5) The rates achieved by the clustering strategy and traditional LFU are plotted against memory M in Fig 0 Clustering performs up to 22 times better than LFU in this example A Caching-and-delivery strategy: memory-sharing V THE MULTI-USER SETUP In this section, we describe the strategy used to achieve the rate approximated by the expression in Theorem 3 Moreover, we will give an exact upper bound on the achieved rate

12 2 We will proceed in two steps First, we will discuss the (H, I, J) partition of the set of levels as first described in Section IV-A This will be accompanied by an explanation of the memory-sharing parameters α i, which indicate what fraction of memory each level gets Second, we analyze the individual rate achieved by every level i, after allocating α i M memory to it, and combine all levels to produce the total rate achieved by the scheme While we actually define the (H, I, J) partition based on the problem parameters, and then choose the α i values accordingly, in this paragraph we will proceed in the opposite order so that we expose the intuition behind the choices As discussed in Section IV-A, the strategy involves finding a good partition (H, I, J) of the set of levels, such that levels in H are given no memory, levels in J are given maximal memory, and levels in I share the rest Thus, for all levels h H, we will assign α h M = 0 Similarly, every level j J will receive α j M = N j /d j, since that is the amount of memory needed to completely store level j and hence to require no BC transmission (ie, R SL = 0; see Theorem ) What is left is to share the remaining memory (M ) j J N j/d j among the levels in I More popular files should get more memory, and the popularity of a level i is proportional to U i /N i Thus, we choose to give level i a memory roughly α i M N i U i /N i (hence the memory per file is proportional to U i /N i ) 7 Intuitively, we want to choose H, I, and J such that the above values of α i are valid, ie, α i [0, ] for all i Based on the intuition, we get the partition described below Definition (M-feasible partition) For any cache memory M, an M-feasible partition (H, I, J) of the set of levels is a partition that satisfies: h H, M < K Ni i I, K U M i ( j J, + ) N j < d j K M, where M = (M T J + V I )/S I, and, for any subset A of the levels: S A = Ni U i ; T A = N i ; d i i A i A U j Nh ; U ( h + d i K V A = i A ) Ni U i ; We stress again that, while we used our own α i values to determine (H, I, J), this was done only for the intuition behind the choice The partition itself is defined solely based on the problem parameters, and not on our strategy Notice in Definition how the different sets are largely determined by the quantity N i /U i, for each level i This matches the idea that the most popular levels (ie, those with the smallest N i /U i ) will be in J, while the least popular levels (those with the largest N i /U i ) will go to the set H After choosing an M-feasible partition, we share the memory among the levels using the following (precise) values of α i : h H, α h M = 0; i I, α i M = N i U i M Ni /K; N i K j J, α j M = N j /d j (7) For completeness, the following proposition states the validity of this choice of memory-sharing parameters Proposition The values of the memory-sharing parameters defined in (7) satisfy: ) α i 0 for all i; 2) i α i = ; 3) α i M N i /d i for all i Note that points and 2 imply α i [0, ] Proof: All three results follow directly from Definition The structure of the solution described above allows us to efficiently compute the α i values Indeed, Algorithm finds α i for all levels i and for all memory values M in Θ(L 2 ) running time, where L is the total number of levels While a detailed description and analysis of the algorithm is given in Appendix A-C, we briefly go over it in this section It proceeds in three main steps In the first step, the algorithm identifies the sequence of (H, I, J) partitions that will occur as M increases, using only the problem parameters For example, if there are two levels, there are two possible sequences, denoted below as S and S2: 7 The square root comes from minimizing an inverse function of α i

13 3 M = 0 large M H {, 2} {2} S I {} {, 2} {2} J {} {, 2} H {, 2} {2} {2} S2 I {} {2} J {} {} {, 2} Notice that the M-feasible partition is the same throughout an entire interval of values for M In fact, there are only 2L non-trivial intervals: each interval is distinguished from the previous one by the promotion of a level from H to I or from I to J As a result, computing only 2L intervals (actually, (2L + 2) intervals, to include the boundary cases) is enough to determine the M-feasible partition for all values of M 0 While Step determines which of the above two sequences (S and S2) will occur, it does not determine the boundaries of the different regimes, ie, the memory values at which the (H, I, J) partition changes (except for trivial boundaries like M = 0) The second step computes these boundaries, using not only the problem parameters, but also the partitions themselves For every M, the corresponding M-feasible partition is determined by the algorithm based on the boundaries calculated in Step 2 For example, suppose the algorithm decided that sequence S should be in use, and that the boundaries of (H, I, J) = (, {, 2}, ) are some values m and m 2 Then, any M [m, m 2 ] has (, {, 2}, ) as its M-feasible partition Algorithm An algorithm that constructs an M-feasible partition for all M Require: Number of caches K and parameters {N i, U i, d i } i for i =,, L Ensure: An M-feasible partition for all M : for all i {,, L} do 2: m i (/K) N i /U i 3: Mi (/d i + /K) N i /U i 4: end for 5: (x,, x 2L ) sort( m,, m L, M,, M L ) 6: 7: Step : Determine (H, I, J) for each interval (x t, x t+ ) 8: Set H 0 {,, L}, I 0, J 0 9: for t {,, 2L} do 0: if x t = m i for some i then : Promote level i from H to I 2: H t H t \ {i} 3: I t I t {i} 4: J t J t 5: else if x t = M i for some i then 6: Promote level i from I to J 7: H t H t 8: I t I t \ {i} 9: J t J t {i} 20: end if 2: end for 22: 23: Step 2: Compute the limits of the intervals as [Y t, Y t+ ) 24: for all t {,, 2L} do 25: Y t x t S It + T Jt V It 26: end for 27: Y 2L+ For convenience 28: 29: Step 3: Determine the M-feasible partition for all M 30: for all t {,, 2L} do 3: Set (H t, I t, J t ) as the M-feasible partition of all M [Y t, Y t+ ) 32: end for The following lemma presents two important properties of M-feasible partitions It will be proved in Appendix A-C Lemma 2 For any cache memory M, an M-feasible partition (H, I, J) always exists Furthermore, the set I is never empty as long as M i N i/d i, ie, as long as individual caches do not have enough memory to store everything

14 4 To properly analyze the achievable rate, we need to look more closely at the set I In the single-level, single-access scenario in [4], [5], three regimes were identified, and they were analyzed separately Generalizing to the single-level, multi-access case, these regimes are: when M < N/K, when M > cn/d for some constant c (0, ), and the intermediate case We identify three similar regimes for each level in i Formally, define: { } 2 Ni I 0 = i I : M < ; K U i { ( β I = i I : M > + ) } Ni ; d i K U i I = I \ (I 0 I ), (8) By choosing the α i values in (7), these definitions are equivalent to: I 0 is the set of levels i such that α i M < N i /K; I is such that α i M > βn i /d i for all i I, where β is the level-separation factor as defined in Regularity Condition (3); and I consists of the remaining levels When K D/β, then I 0, I, and I are mutually exclusive and form a partition of I For convenience, we call the resulting partition (H, I 0, I, I, J) a refined M-feasible partition 8 What follows is an important statement regarding the set I Proposition 2 (Size of I ) In any refined M-feasible partition (H, I 0, I, I, J) as defined in Definition and (8), the set I contains at most one element Proof: Suppose that there exist two level i, j I (and possibly others) We will show that this violates regularity condition (3) Suppose without loss of generality that i is more popular than j Since i, j I, then, by Definition and (8): ( β + ) ( N j < M + ) Ni d j K d i K U i However, this means: U j U i /N i U j /N j < /d i + /K β/d j + /K < d j βd i D β, which contradicts regularity condition (3) Using the definition of a refined M-feasible partition, and the values of {α i } i, we give upper bounds on the rates achieved individually for each level Lemma 3 Given a refined M-feasible partition (H, I 0, I, I, J), the individual rates of the levels are upper-bounded by: h H, R h (M) = KU h ; i I 0 I, R i (M) 2S I Ni U i ; M T J + V I i I, R i (M) ( β d i U i M T ) J N i /d i j J, R j (M) = 0 The total achieved rate is then: R(M) = L i= R i(m) + β d S I0 + S I i U i ; Ni U i We relegate the proof of Lemma 3 to Appendix A-C What follows is a brief explanation of the individual rates Since levels h H get no memory, no information about their files can be stored in the caches Hence, the server will have to transmit a complete copy of every file requested by the KU h users in level h Therefore, R h (M) = KU h in the worst case Since levels j J get maximal memory, any file requested from j can be fully recovered using the caches Thus there is no need for the server to send anything for j, and hence R j (M) = 0 Finally, the levels i I get enough of the remaining memory M T J so that they behave as in Theorem However, since α i M N i U i /S I (M T J ), we get a rate of: R i (M) N iu i α i M d iu i S I Ni U i d i U i M T J 8 The fact that this is a partition is important for the case K D/β In the opposite case, this does not necessarily hold but it will not matter anyway